Approximating the Longest Path Length of a Stochastic DAG by a Normal Distribution in Linear Time ∗ Ei Ando A, , Toshio Nakata B, Masafumi Yamashita A,C
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Discrete Algorithms 7 (2009) 420–438 Contents lists available at ScienceDirect Journal of Discrete Algorithms www.elsevier.com/locate/jda Approximating the longest path length of a stochastic DAG by a normal distribution in linear time ∗ Ei Ando a, , Toshio Nakata b, Masafumi Yamashita a,c a Dept. Computer Sci. and Communication Eng., Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan b Dept. of Mathematics, Fukuoka University of Education, Akama-Bunkyomachi, Munakata, Fukuoka, 811-4192, Japan c Institute of Systems, Information Technologies and Nanotechnologies, Fukuoka SRP Center Building 7F 2-1-22, Momochihama, Sawara-ku, Fukuoka, 814-0001, Japan article info abstract Article history: This paper presents a linear time algorithm for approximating, in the sense below, the Received 3 September 2007 longest path length of a given directed acyclic graph (DAG), where each edge length is Received in revised form 7 January 2009 given as a normally distributed random variable. Let F (x) be the distribution function of Accepted 12 January 2009 the longest path length of the DAG. Our algorithm computes the mean and the variance Available online 21 January 2009 of a normal distribution whose distribution function F˜ (x) satisfies F˜ (x) F (x) as long as Keywords: F (x) a,givenaconstanta (1/2 a < 1). In other words, it computes an upper bound ˜ −1 Directed acyclic graph 1 − F (x) on the tail probability 1 − F (x),providedx F (a). To evaluate the accuracy ˜ Longest path problem of the approximation of F (x) by F (x), we first conduct two experiments using a standard Stochastic edge weight benchmark set ITC’99 of logical circuits, since a typical application of the algorithm is the Normal distribution delay analysis of logical circuits.
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